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Institute of Information Theory and Automation Introduction to Pattern Recognition Jan Flusser

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Recognition (classification) = assigning a pattern/object to one of pre-defined classes Pattern Recognition

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Recognition (classification) = assigning a pattern/object to one of pre-defined classes Statictical (feature-based) PR - the pattern is described by features (n-D vector in a metric space) Pattern Recognition

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Recognition (classification) = assigning a pattern/object to one of pre-defined classes Syntactic (structural) PR - the pattern is described by its structure. Formal language theory (class = language, pattern = word) Pattern Recognition

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Supervised PR – training set available for each class Pattern Recognition

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Supervised PR – training set available for each class Unsupervised PR (clustering) – training set not available, No. of classes may not be known Pattern Recognition

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PR system - Training stage Definition of the features Selection of the training set Computing the features of the training set Classification rule setup

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Desirable properties of the features Invariance Discriminability Robustness Efficiency, independence, completeness

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Desirable properties of the training set It should contain typical representatives of each class including intra-class variations Reliable and large enough Should be selected by domain experts

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Classification rule setup Equivalent to a partitioning of the feature space Independent of the particular application

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PR system – Recognition stage Image acquisition Preprocessing Object detection Computing of the features Classification Class label

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An example – Fish classification

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The features: Length, width, brightness

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2-D feature space

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Empirical observation For a given training set, we can have several classifiers (several partitioning of the feature space)

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Empirical observation For a given training set, we can have several classifiers (several partitioning of the feature space) The training samples are not always classified correctly

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Empirical observation For a given training set, we can have several classifiers (several partitioning of the feature space) The training samples are not always classified correctly We should avoid overtraining of the classifier

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Formal definition of the classifier Each class is characterized by its discriminant function g(x) Classification = maximization of g(x) Assign x to class i iff Discriminant functions defines decision boundaries in the feature space

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Minimum distance (NN) classifier Discriminant function g(x) = 1/ dist(x, ) Various definitions of dist(x One-element training set

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Voronoi polygons

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Minimum distance (NN) classifier Discriminant function g(x) = 1/ dist(x, ) Various definitions of dist(x One-element training set Voronoi pol NN classifier may not be linear NN classifier is sensitive to outliers k-NN classifier

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Find nearest training points unless k samples belonging to one class is reached

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Discriminant functions g(x) are hyperplanes Linear classifier

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Assumption: feature values are random variables Statistic classifier, the decission is probabilistic It is based on the Bayes rule Bayesian classifier

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The Bayes rule A posteriori probability Class-conditional probability A priori probability Total probability

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Bayesian classifier Main idea: maximize posterior probability Since it is hard to do directly, we rather maximize In case of equal priors, we maximize only

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Equivalent formulation in terms of discriminat functions

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How to estimate ? Parametric estimate (assuming pdf is of known form, e.g. Gaussian) Non-parametric estimate (pdf is unknown or too complex) From the case studies performed before (OCR, speech recognition) From the occurence in the training set Assumption of equal priors

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Parametric estimate of Gaussian

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d-dimensional Gaussian pdf

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The role of covariance matrix

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Two-class Gaussian case in 2D Classification = comparison of two Gaussians

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Two-class Gaussian case – Equal cov. mat. Linear decision boundary

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Equal priors Classification by minimum Mahalanobis distance If the cov. mat. is diagonal with equal variances then we get “standard” minimum distance rule max min

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Non-equal priors Linear decision boundary still preserved

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General G case in 2D Decision boundary is a hyperquadric

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General G case in 3D Decision boundary is a hyperquadric

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More classes, Gaussian case in 2D

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What to do if the classes are not normally distributed? Gaussian mixtures Parametric estimation of some other pdf Non-parametric estimation

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Non-parametric estimation – Parzen window

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The role of the window size Small window overtaining Large window data smoothing Continuous data the size does not matter

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n = 1 n = 10 n = 100 n = ∞

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The role of the window size Small window Large window

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Applications of Bayesian classifier in multispectral remote sensing Objects = pixels Features = pixel values in the spectral bands (from 4 to several hundreds) Training set – selected manually by means of thematic maps (GIS), and on-site observation Number of classes – typicaly from 2 to 16

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Satellite MS image

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Other classification methods in RS Context-based classifiers Shape and textural features Post-classification filtering Spectral pixel unmixing

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Typically for “YES – NO” features Feature metric is not explicitely defined Decision trees Non-metric classifiers

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General decision tree

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Binary decision tree Any decision tree can be replaced by a binary tree

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Real-valued features Node decisions are in form of inequalities Training = setting their parameters Simple inequalities stepwise decision boundary

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Stepwise decision boundary

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Real-valued features The tree structure and the form of inequalities influence both performance and speed.

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How to evaluate the performance of the classifiers? - evaluation on the training set (optimistic error estimate) - evaluation on the test set (pesimistic error estimate) Classification performance

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How to increase the performance? - other features - more features (dangerous – curse of dimensionality!) - other (larger, better) traning sets - other parametric model - other classifier - combining different classifiers Classification performance

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Combining (fusing) classifiers C feature vectors Bayes rule: max max Several possibilities how to do that

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Product rule Assumption: Conditional independence of x j max

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Max-max and max-min rules Assumption: Equal priors max (max ) max (min )

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Majority vote The most straightforward fusion method Can be used for all types of classifiers

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Training set is not available, No. of classes may not be a priori known Unsupervised Classification (Cluster analysis)

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What are clusters? Intuitive meaning - compact, well-separated subsets Formal definition - any partition of the data into disjoint subsets

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What are clusters?

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How to compare different clusterings? Variance measure J should be minimized Drawback – only clusterings with the same N can be compared. Global minimum J = 0 is reached in the degenerated case.

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Clustering techniques Iterative methods - typically if N is given Hierarchical methods - typically if N is unknown Other methods - sequential, graph-based, branch & bound, fuzzy, genetic, etc.

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Sequential clustering N may be unknown Very fast but not very good Each point is considered only once Idea: a new point is either added to an existing cluster or it forms a new cluster. The decision is based on the user-defined distance threshold.

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Sequential clustering Drawbacks: -Dependence on the distance threshold -Dependence on the order of data points

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Iterative clustering methods N-means clustering Iterative minimization of J ISODATA Iterative Self-Organizing DATa Analysis

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N-means clustering 1.Select N initial cluster centroids.

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N-means clustering 2. Classify every point x according to minimum distance.

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N-means clustering 3. Recalculate the cluster centroids.

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N-means clustering 4. If the centroids did not change then STOP else GOTO 2.

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N-means clustering Drawbacks - The result depends on the initialization. - J is not minimized - The results are sometimes “intuitively wrong”.

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N-means clustering – An example Two features, four points, two clusters (N = 2) Different initializations different clusterings

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Initial centroids N-means clustering – An example

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Initial centroids N-means clustering – An example

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Iterative minimization of J 1.Let’s have an initial clustering (by N-means) 2.For every point x do the following: Move x from its current cluster to another cluster, such that the decrease of J is maximized. 3. If all data points do not move, then STOP.

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Example of “wrong” result

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ISODATA Iterative clustering, N may vary. Sophisticated method, a part of many statistical software systems. Postprocessing after each iteration -Clusters with few elements are cancelled -Clusters with big variance are divided -Other merging and splitting strategies can be implemented

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Hierarchical clustering methods Agglomerative clustering Divisive clustering

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Basic agglomerative clustering 1.Each point = one cluster

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Basic agglomerative clustering 1.Each point = one cluster 2.Find two “nearest” or “most similar” clusters and merge them together

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Basic agglomerative clustering 1.Each point = one cluster 2.Find two “nearest” or “most similar” clusters and merge them together 3.Repeat 2 until the stop constraint is reached

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Basic agglomerative clustering Particular implementations of this method differ from each other by - The STOP constraints - The distance/similarity measures used

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Simple between-cluster distance measures d(A,B) = d(m 1,m 2 ) d(A,B) = min d(a,b) d(A,B) = max d(a,b)

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Other between-cluster distance measures d(A,B) = Hausdorf distance H(A,B) d(A,B) = J(AUB) – J(A,B)

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Agglomerative clustering – representation by a clustering tree (dendrogram)

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How many clusters are there? 2 or 4 ? Clustering is a very subjective task

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How many clusters are there? Difficult to answer even for humans “Clustering tendency” Hierarchical methods – N can be estimated from the complete dendrogram The methods minimizing a cost function – N can be estimated from “knees” in J-N graph

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Life time of the clusters Optimal number of clusters = 4

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Optimal number of clusters

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Applications of clustering in image proc. Segmentation – clustering in color space Preliminary classification of multispectral images Clustering in parametric space – RANSAC, image registration and matching Numerous applications are outside image processing area

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References Duda, Hart, Stork: Pattern Clasification, 2 nd ed., Wiley Interscience, 2001 Theodoridis, Koutrombas: Pattern Recognition, 2 nd ed., Elsevier, 2003

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